3. Homogenized problem#
3.1. Homogenized problem obtained#
To take into account the periodic nature of the medium studied, a homogenization method is used based in this specific case on an asymptotic development of the variables involved in the initial physical problem. With regard to the operating procedure, the reader is referred to the following references [bib2], [bib4], [bib5], [bib6]. Here, we will simply state the results obtained.
In homogenized medium \({\Omega }_{0}\) (see [fig 3.1-a]), the following two homogenized variables are considered: \({s}_{0}\) (movement of the beams) and \({\Phi }_{0}\) (fluid displacement potential). In variational form, these variables are linked by the natural vibration equations:
\(\mathrm{\{}\begin{array}{cc}{\mathrm{\int }}_{{\Omega }_{0}}\mathrm{A}\mathrm{.}\mathrm{\nabla }{\Phi }_{0}\mathrm{.}\mathrm{\nabla }{\phi }^{\text{*}}\mathrm{=}\mathrm{-}{\mathrm{\int }}_{{\Omega }_{0}}\mathrm{D}\mathrm{.}{\mathrm{s}}_{0}\mathrm{.}{\phi }^{\text{*}}& \mathrm{\forall }{\phi }^{\text{*}}\mathrm{\in }{V}_{\Phi }^{\text{hom}}\\ {\mathrm{\int }}_{{\Omega }_{0}}\mathrm{M}\mathrm{.}\frac{{\mathrm{\partial }}^{2}{\mathrm{s}}_{0}}{\mathrm{\partial }{t}^{2}}{\mathrm{s}}^{\text{*}}+{\mathrm{\int }}_{{\Omega }_{0}}\mathrm{K}\mathrm{.}\frac{{\mathrm{\partial }}^{2}{\mathrm{s}}_{0}}{\mathrm{\partial }{z}^{2}}\frac{{\mathrm{\partial }}^{2}{\mathrm{s}}^{\text{*}}}{\mathrm{\partial }{z}^{2}}\text{=}{\rho }_{F}{\mathrm{\int }}_{{\Omega }_{0}}\mathrm{D}\mathrm{.}\mathrm{\nabla }\frac{{\mathrm{\partial }}^{2}{\Phi }_{0}}{\mathrm{\partial }{t}^{2}}\mathrm{.}{\mathrm{s}}^{\text{*}}& \mathrm{\forall }{\mathrm{s}}^{\text{*}}\mathrm{\in }{V}_{s}^{\text{hom}}\end{array}\) eq 3.1-1
where:
\(\begin{array}{c}{V}_{s}^{\text{hom}}\mathrm{=}{L}^{2}{({\Omega }_{0})}^{2}\mathrm{\times }{H}_{0}^{2}{({\Omega }_{0})}^{2}\\ \text{où}{\Omega }_{0}\mathrm{=}S\mathrm{\times }\mathrm{]}\mathrm{0,}L\mathrm{[}\\ {H}_{0}^{2}({\Omega }_{0})\mathrm{=}\left\{v;\mathrm{\forall }(x,y)\mathrm{\in }Sz\to v(x,y,z)\mathrm{\in }{H}_{0}^{2}(\mathrm{]}\mathrm{0,}L\mathrm{[})\right\}\end{array}\)
.
Figure 3.1-a
Figure 3.1-b
The various tensors that are involved in [éq 3.1-1] are defined using two functions in \({\chi }_{\alpha }(\alpha \mathrm{=}\mathrm{1,2})\) as follows:
\(B=({b}_{\mathrm{ij}})=\frac{1}{∣Y∣}(\begin{array}{ccc}{\int }_{{Y}_{F}}\frac{\partial {\chi }_{1}}{\partial {y}_{1}}& {\int }_{{Y}_{F}}\frac{\partial {\chi }_{1}}{\partial {y}_{2}}& 0\\ {\int }_{{Y}_{F}}\frac{\partial {\chi }_{2}}{\partial {y}_{1}}& {\int }_{{Y}_{F}}\frac{\partial {\chi }_{2}}{\partial {y}_{2}}& 0\\ 0& 0& 0\end{array})A=({a}_{\mathrm{ij}})=(\frac{∣{Y}_{F}∣}{∣Y∣}{\delta }_{\mathrm{ij}}-{b}_{\mathrm{ij}})\) eq 3.1-2
\(\begin{array}{}D=({d}_{\mathrm{ij}})=B+\frac{1}{∣Y∣}(\begin{array}{ccc}{Y}_{S}& 0& 0\\ 0& {Y}_{S}& 0\\ 0& 0& 0\end{array})M=({m}_{\mathrm{ij}})={\rho }_{F}B+\frac{{\mu }^{2}}{∣Y∣}(\begin{array}{ccc}{\rho }_{S}S& 0& 0\\ 0& {\rho }_{S}S& 0\\ 0& 0& 0\end{array})\\ K=({k}_{\mathrm{ij}})=\frac{E{\mu }^{2}}{∣Y∣}(\begin{array}{ccc}{I}_{\mathrm{xx}}& {I}_{\mathrm{xy}}& 0\\ {I}_{\mathrm{xy}}& {I}_{\mathrm{yy}}& 0\\ 0& 0& 0\end{array})\end{array}\) eq 3.1-3
where \(∣{Y}_{F}∣\) and \(∣{Y}_{S}∣\) represent respectively the areas of the fluid and structure domains of the elementary reference cell (cf. [fig 3.1-b]). \(∣Y∣\) represents the sum of the previous two areas. The elementary reference cell is homothetic with a ratio of \(\mu\) to the real periodicity cell of the heterogeneous medium.
The two \({\chi }_{\alpha }(\alpha =\mathrm{1,2})\) functions are solutions to a two-dimensional problem, called a cellular problem. On the elementary reference cell, functions \({\chi }_{\alpha }(\alpha =\mathrm{1,2})\) are defined by:
\(\begin{array}{}{\int }_{{Y}_{F}}\nabla {\chi }_{\alpha }\mathrm{.}\nabla v={\int }_{\gamma }{n}_{\alpha }\mathrm{.}v\forall v\in V\\ {\int }_{{Y}_{F}}{\chi }_{\alpha }=0(\text{pour avoir une solution unique})\end{array}\) eq 3.1-4
where:
\(V=\text{{}v\in {H}_{1}({Y}_{F}),v(y)\text{périodique en}y\text{de période 1}\text{}}\)
Note:
It is shown that the two-dimensional part of \(B\) is symmetric and positive definite [bib5].
Note:
In the matrix \(M\) , the term \({\rho }_{F}\mathrm{B}\) plays the role of an added mass matrix specific to each beam in its cell.
Note:
For the different tensors, we can factor the multiplicative term \(\frac{1}{∣Y∣}\). It was added in order to obtain the « right mass » of tubes in the absence of fluid. We then \({\int }_{{\Omega }_{0}}M\text{dV}\) = mass of the tubes composing \({\Omega }_{0}\) .
3.2. Matrix problem#
By discretizing the problem [éq 3.1-1] by finite elements, we end up (with obvious notations) with the following problem:
\(\{\begin{array}{}\stackrel{ˆ}{A}{\Phi }_{0}=-\stackrel{ˆ}{D}{s}_{0}\\ \stackrel{ˆ}{M}\frac{{\partial }^{2}{s}_{0}}{\partial {t}^{2}}+\stackrel{ˆ}{K}{s}_{0}={\rho }_{F}{\stackrel{ˆ}{D}}^{T}\frac{{\partial }^{2}{\Phi }_{0}}{\partial {t}^{2}}\end{array}\) eq 3.2-1
This can be put in the form (we pre-multiply the first equation by \({\rho }_{F}\)):
\(\tilde{M}(\begin{array}{}\frac{{\partial }^{2}{s}_{0}}{\partial {t}^{2}}\\ \frac{{\partial }^{2}{\Phi }_{0}}{\partial {t}^{2}}\end{array})+\tilde{K}(\begin{array}{}{s}_{0}\\ {\Phi }_{0}\end{array})=(\begin{array}{cc}\stackrel{ˆ}{M}& -{\rho }_{F}{\stackrel{ˆ}{D}}^{T}\\ -{\rho }_{F}\stackrel{ˆ}{D}& -{\rho }_{F}\stackrel{ˆ}{A}\end{array})(\begin{array}{}\frac{{\partial }^{2}{s}_{0}}{\partial {t}^{2}}\\ \frac{{\partial }^{2}{\Phi }_{0}}{\partial {t}^{2}}\end{array})+(\begin{array}{cc}\stackrel{ˆ}{K}& 0\\ 0& 0\end{array})(\begin{array}{}{s}_{0}\\ {\Phi }_{0}\end{array})=0\) eq 3.2-2
Note:
The problem obtained is symmetric. If instead of choosing the displacement potential to represent the fluid, we had chosen the speed potential, we would have obtained a non-symmetric matrix problem also showing a damping matrix.
Note:
It is necessary to have \({\rho }_{F}>0\) in order for the mass matrix \(\tilde{\mathrm{M}}\) to be invertible. If you want to do a calculation in « air », cf. § 6.3 , you have to block degrees of freedom in \({\Phi }_{0}\) .